Elliptic classes of Schubert varieties via Bott–Samelson resolution

Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized full flag variety G / B . For this first we need to twist the notion of elliptic characteristic class of Borisov–Libgober by a line bundle, and thus allow the elliptic classes to depend on extra variables. Using the Bott–Samelson resolution of Schubert varieties we prove a BGG-type recursion for the elliptic classes, and study the Hecke algebra of our elliptic BGG operators. For G = GL n ( C ) we find representatives of the elliptic classes of Schubert varieties in natural presentations of the K theory ring of G / B , and identify them with the Tarasov–Varchenko weight function. As a byproduct we find another recursion, different from the known R-matrix recursion for the fixed point restrictions of weight functions. On the other hand the R-matrix recursion generalizes for arbitrary reductive group G.